Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*)
Progress of Thc\.retical Physics
Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*>
Hiromichi EBISAWA 0 1
Hidetoshi FUKUYAMA 0 1
0 Department of Physics, University of Tokyo , Tokyo
1 > Part of Ph. D. Thesis submitted by H. Ebisawa to University of Tokyo , 1970
The time dependent Ginzburg Landau equation and the fluctuation propagator of a superconductor in a static electromagnetic field are examined to the next order of the small parameters 1/SF't" and TfsF, where sF is the Fermi energy and 't" the relaxation time. The coefficient of the time derivative of the order parameter becomes a complex number, whose imaginary part is of order of TlsF smaller than the real part. This term is shown to be important for Hall effect due to the fluctuation near Tc, which has not been expected in the usual approximation. The arguments cover the cases of arbitrary mean free paths near Tc and of arbitrary temperatures except T<f:.Tc.

Wave Character of the Time Dependent Ginzburg Landau Equation 1043
Integration over a momentum variable is transformed into the energy integral
multiplied by the density of states, N, at the Fermi energy.
By these approximations, we have
with
ID (q, w +. io) =  N_!_ [77 ilow + Aq2]\
f) . 1
r?Jf (r, t) =  P2?JI' (r, t)  a?JI' (r, t)'
fJt 2m
A=nD
8T'
a=~7J2mtt'
r = 2~tt,
where v and r are the Fermi velocity and relaxation time, respectively. Thus
all coefficients are real positive numbers. In Eqs. (
1
) and (
2
), we neglect terms
of the order of 1/sFr and T/sF, where SF is the Fermi energy. Using ID, Eq.
(
1
), or the TDGL equation, Eq. (
2
), we can examine such dynamical properties
as electrical resistivity and the ultrasonic attenuation. However as regards the
Hall effect, Caroli and Maki3> obtained a vanishing contribution in the vortex
state, and Tsuzuki and the present authors6> showed that the Hall conductivity
does not have contributions from the AL process slightly above Tc. These van
ishing results on Hall effects are intimately connected with the fact that Ao and
A in Eq. (
1
) or Eq. (
2
) are real quantities. The reality of Ao and A are, however,
true only if we neglect terms of the order of 1/sFr and T/sF in the derivation
of ID. Thus we are in need to determine the fluctuation propagator ID and the
TDGL equation to this order to examine some dynamical processes. These de
terminations up to this order in the presence of external electric and magnetic
fields are the purpose of the present article. We confine ourselves within the
Born approximations and the ladder corrections to impurity scattering. That is,
we work within (i) and (ii), but need not (iii) and (iv).
In § 1 the derivation is given for the nearly free electron system to clarify
the expansion parameters. As is shown the corrections, which is of order of
T /sF, depend on the energy dependence of the density of states function near
the Fermi energy. Then we consider in § 3 the cases of arbitrary Bloch electrons.
In § 4 we briefly discuss a new effect due to these corrections.
§ 2. Fluctuation propagator and TDGL equation for a
nearly free electron system
For nearly free electrons, the structure of the propagator g) is very simple.
The model Hamiltonian is
(
1
)
(
2
)
(
3
)
(
4
)
where
1 .
=2nniu2N,
r
N is the density of states
= 1 +X   lr + ... ,
2sF 2
N= mkF.
2n2
u is the Fourier transform of U(r). · The branch of square root is such as Im <f?1R>O.
From now on, the approximation, Re 1: = 0, is adopted.
In the ladder approximation for the BCS coupling, the fluctuation propagator
Is given by
1 i[i i[i
Il(q, iw;..) = 
{3 0 0
where
drdr' exp[iw;..(rr')J<T..P"t(q, r)P'( q, r')), (
13
)
(
6
)
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
(
12
)
(14)
Here IJR is an analytic continuation of II(q, iw;..) from the upper plane of{)) and
the bracket means the thermal average. Adopting the ladder approximation for
impurity scattering, which is consistent with Eq. (
8
), one obtains
IJR(q, w) =~fro dx tanh _!E_[flRR(x, x+ w) flAR(x, x+w)
4nz oo 2T
+flAR(xw, x) flAA(x()), x)],
(15)
where
in the small q limit, where
C(J2R(x')== j 1+
x' .SA( x')
e]i'
x' i
_1+2e]i'+2r+···.
(19)
One can immediately see that the fourth term in Eq. (15) gives the same con
tribution as the first. In the dirty limit Eq. (15) is calculated as
=N (1 +w )\ sco dxtanhx
2 4eF co 2T
x {[{x+
2
w+iDq (
2
w )}1
1+ 4c;F
. _1]
 (x+zo)
1 }
+ x+io '
which is valid to the linear order of w. Here we have performed the expansiOn
of cp's with respect to 1/c;Fr and x/c;Fr...JT/c;F (or WD/c;F) to the order following
the one that leads to the ordinary ID. The expansion parameter with q2 is rDq2•
Introducing the BCS cutoff, !xi <wD, in the second term, and using the relation
l__JIR(O, 0) = l_Nln 2rwD =ln_Z_,
g g nT Teo
one obtains JIR and then IIJR as follows:
[I!JR(q,w)]1=N{ln ~o +w(~ +t:)w(~)
where ¢ 1s the digamma function and
+_!_[_!_ln_Z_ w(l_ + e:) + ¢ (l_)]}'
4c;F gN Teo 2 2
(=
 iw + Dq 2 (
47C T
w )
1 +4eli', ·
(20)
(21)
(22)
where X(p) is the Gor'kov function, B) given by
X(p) = _8_1_ {n2 + l_[¢ (_!_)  ¢(_!_ + _!_P)} J
7( (
3
) p 8 2p 2 2 2 '
1
p= 2nrT·
So far we have neglected the effects of electric fields. For the derivation
of the TDGL equation which includes a scalar potential of an electric field, careful
treatments are necessary concerning the ordering of the operator 8/8r and the
potential. Introducing an external field by
In the presence of a magnetic field, q should be replaced by Q = q 2eA.7> This
result, Eq. (23), has new terms proportional to w, which is smaller than the
ordinary terms by a factor T / cp.
If Dq2/T<1, we get
[.fDR(q, w)]1= N[ln T +J...q2 iJ...0w+_!!!__1]~
Teo 4sFgN
This equation is valid for cases with arbitrary values of mean free path, if we
set
7( (
3
)v2
J... 48(nT)2 x(p),
one has the equation for the order parameter Jt,9>
to the linear order of V.
The function P is defined by
P(qh q2, iw}o) =  _!_ fP fP drdr' exp [iw}o (r r')]
{3 Jo Jo
Jt(q, w) = g {JIR(q, w) Jt (q, w) + PR(qh q2, w) Jot Cq1) 2e V(q2, w)},
where
x (T./Jft Cq1 + q2, r) 1Jf ( qh r) n ( q2, r'))
n(q) =
Jdr exp[ iq·r]¢tt(r)¢t(r).
In the dirty limit, a similar calculation to those for JIR yield PR(qh q2, w) for
the limit, w___,.O (Appendix A).
iN Dq2 )
PR(qt, q 2) =  4nT ¢<1>(21 + 4nT
2
N [ 1 T ( 1 Dq1 ) (
1
)
+ 4cp gNln Teo¢ 2+ 4nT +¢ 2
(24)
(25)
(26)
(27)
(28)
(29)
Wave Character of the Time Dependent Ginzburg Landau Equation 1047
4rcTn~DQ2 fL2eV(r), 2sn +1DQ2 J2sn +1DQ2 }Jt(r,t) =O,
where [ , ] is a commutator and
Q=q2eA.
§ 3. Fluctuation propagator in the case of Bloch electrons
We assume an arbitrary Sk k relation in a single band dispersion, restricting
ourselves only to systems with the cubic symmetry. In this section we confine
ourselves only in the dirty limit, and in the systems with a Fermi energy such
that 1/sFr<1, T/sF<1 and (})D/sF<l. Equation (15) is still valid if one use the
correct expressions for fl, i.e., for g. The selfenergy is calculated as
where N(x) is the density of states at cp+ x. Similarly one obtains
gAR(x,x)= JdskN(sk)
1 . 1 .
xsk+Spz/2r xsk+sFz/2r
~ircN',
where N' is the derivative of N with respect to x at x = 0.
Using the relation
and neglecting the quantity of the order of (})D/cp, one gets
JIR, 2+3 (q, (})) = (}). SIJJD dx tanhX  d[gAR(x, x) 1 niu2] 1 =0.
4rcz IJ)n 2T dx
(34)
Another branch of g is expressed as
gRR(x, x+(})) =~ GR(k, x)GA(k, x(}))
k
(32)
(33)
where the first term is expanded in terms of (J) as
The k summation can be performed if one notes that one can expand the de
nominator of G in terms of rx/"'.JrT.
· (
:E GR(k,x)GA(k, x) = SdeN(e) [eFe+~z~x 1+~z·~~N')]
k 2r 2r N
1
X [Spe z·~+x (1z·__N_'
)]2r 2r N
1
=2rcNr(12rx),
:E GR(k, x)_LGA(k, x{f)) = :E _J__GR(k, x)_J__GA(k, x(J))
k 8k/ k f)kp f)kp
= :E GR(k, x)GA(k, x(J)Y(_J__e(k))
.k f)~
= :E(_]_e(k) rGR(k, x)2GA(k, x)2
k f)kp
2
2
2{)) (1i_ N' ):E(_J__e(k)) GR(k, x)2GA(k, x)3•
2r N k f)kp
The first of Eq. (39) is evaluated as follows.
 :E(1__e(k) rGR(k, x)2GA(k, x)2
k f)k,.
· ( · N' ).  2
= Sdea(e) [Spe+ dr x 1+ dr N J
X [SpSz·+x ( 1 z·N~~')]
2r 2r N
2
=4rcr3a(sF) (1+3irx),
where
(40) one obtains
:E(_]__s(k) rGR(k, x)2GA(k, x)S
k f)k,.
(37)
(38)
(39)
(40)
(41)
gRR(x,x+U))=2nN{1+iU)r+2irx[1+iU)r(1 1r ~)]}2rcr3aq2•
Thus we get where
JI1+ 4 (q, 0)) =~ Jdx tanh ~[gRR(x, x + 0))1  niu2]
2nz 2T
=N[1 +2iU)r (1_i__ N' )] fdx tanh~ 1
2 4r N 2 T x + i( '
1
( =   ( iU) +Dq2) [1 + 2iU)r (1  _i_ N' ) ] ,
4nT 4r N
D=ar2/N.
+0) N'[ 1 <J; (1+() +<J; (1)' lnT~J} .
2 N .gN 2 2 Te
§ 4. Discussion
The same procedures of the integration over x as used In § 2 yield
Again q should be replaced by Q = q 2eA in the presence of a magnetic field.
We have derived the fluctuation propagator and the TDGL equation for the
order parameter and found the new terms proportional to 0) of order of TjsF.
Though it is small it is important in some situation. One can show that there
is a contribution to Hall conductivity due to fluctuation by use of the newly
derived TDGL equation, Eq. (31), instead of Eq. (
2
). The arguments are restrict
ed in the range near Te. The basic equations are
h(r+is)( :t +2ie¢)Jt(r, t) = [ 2~( ir 2:Ar+a]Jt(r, t),
j(r, t) = i!__(y P' +4i~A)Jt(r', t)A(r, t) I ,
2m c ~~
where
e=2ar.
7C
(42)
(43)
(44)
(45)
(46)
(47)
(48)
As we are concerned only with weak magnetic field cases, we apply the Wigner
representation10> to treat the magnetic field for the system of fluctuating order
parameters. Using the formal solution Jt (t), one gets the density matrix of the
system as
where
p=(J(t)Jt(t))
=2kBT st dt' exp [(
hr co
h(r is)
!}[
h(r+ ie)
i~ext ) (t t') J•1
+ i!J{ext ) (t _ t') J
h '
!J[ =1'It2 +a,
2m
1t = p +2.!!.__ A ,
c
!}[ext= 2eEx .
The bracket means ensemble average with respect to stochastic variables, and 1
means the unit matrix. The factor in front of the integral in Eq. (50) is adjusted
such that in the absence of external fields p may be consistent with GL free
energy.
Constructing the Liouville operator for the Wigner distribution function I
corresponding to p from the equation of motion, one obtains
fi/ = i (.£o + .£' + Lext) I+ 2kBT '
fit hr
i.£ =  2 ( 1n2 +a)  .£ !!_ _}_,
hr 2m /!. m fix
l·.,Lr' _  e2(J)c ( 7Cfxi7C yfi )
r2 finy finx '
i.£ext = 2eE_}_.
finx
fD=
kBT
(1/2m) 7t2 +a
The equilibrium distribution function IS
Following Kubo's derivation of the expression for the conductivity tensor to the
linear order of H, one gets
(50)
(51)
(52)
(53)
(54)
Wave Character of the Time Dependent Ginzburg Landau Equation 1051
where the restriction of phase integral is used. Explicit calculations lead to
The temperature dependence is thus found more singular than (J~x by Aslamazov
Larkin.1)
The full discussion of this large contribution to the Hall conductivity due
to fluctuations will be given in a following paper, based on the microscopic theory,
including contributions form the Maki process.11)
A note is added concerning the difference of the new term in the fluctuation
propagator or the TDGL equation form the corresponding one by Abrahams
Tsuneto12) and Maki.18) Theirs include the q2 term as
( D+ 4_mi)q2.
As is shown in Appendix B, g)R (q, w) is a real number for w= 0 in any approxi
mation for the impurity scattering other than the Born and the ladder one as far
as we work in the ladder approximation for the BCS coupling.
In such approximations we may expect the corrections of order of 1/c.ur to
the coefficient of w, which do not appear in the ladder approximation for the
impurity scattering. The consistent treatments over 1: and the vertex corrections
must be done. One more open problem now is the validity of neglecting the
real part of 1:, which is closely related to the model potential due to impurities.
Acknowledgements
The authors thank Professor T. Tsuzuki, Professor K. Maki, Professor R.
Kubo and Professor Y. W ada for useful discussions. They are grateful for
financial support by the Research Institute for Fundamental Physics, Kyoto U niver
sity. One of the authors (H.F.) is thankful to the Sakkokai Foundations for
financial support.
Appendix A
Calculation of pR (q, q')
In the ladder approximation Eq. (29) becomes
(58)
(59)
X [1 niu2g(qh isn, isn) ]1 [1 niu2g(q2, isn,  isn + iwA) ]1
X [1 niu2g(q1 + q2, isn, isn iwA) ]1•
(A·1)
The analytically continued function of the summand is calculated similarly to fjBB',
as
1
X(/}11+(/)2irCq1+q2)2/3kF2·ir1 (cp/+cp2)
2 (<Pl + <P.'~ (<P.' + <P•) J
+ (q1 +kFq22)2 (terms obtam.ed by exchange of cp1 and cp/) },
3
where cp~, cp2 or cp/ means the abbreviation of cp1B (x), cp2B (x) or cp1B' (x'), respec
tively. Here q'2 is put to zero because the space charge, P2V, is vanishing.
Expanding cp's in terms of T/sF and 1/sFr, one obtains, to the lowest order of
{)),
P R Cq1 q2) =
' 4
dx t a n2XTh2XsF x+iD1/2·q12 x+iD/21· (q1+q2)2
+_wN_Jdxtanhx__d_[ 2 1 _1_(1+iD(q1+q2Y)]
4 2T dx w(1x/2sF) xiD/2·q12 2 4sF
'
(A·2)
which leads to Eq. (30).
Reality of ilJR(q, 0)
By definition, we have
IJR(q, 0) = ~ fdx rdrexp[iq(rr')]
4m J
Appendix B
where bar means the ensemble average over the random distribution of impuri
ties. In Eq. (B ·1) we have put {)) = 0, for no singularity appears if (J))0.
Impurity potential is real and the scattering is elastic.
Thus JIR(q, 0) 1s real.
[GR(r, r', x)]*=GA(r, r', x).
D
o
w
n
l
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
p
t
p
.
o
x
f
o
r
d
j
o
u
r
n
a
.l
s
o
r
g
/
b
y
g
u
e
s
t
o
n
J
u
n
e
2
1
,
2
0
1
6
1) L. G. Aslamazov and A. I. Larkin , Phys. Letters 26A ( 1968 ), 238 ; Fiz. Tverd. Tela 10 ( 1968 ), 1104 [ Soviet Phys . Solid State 10 ( 1968 ), 875 ].
2) H. Schmidt , z. Phys . 216 ( 1968 ) , 336 .
A. Schmid , z. Phys . 215 ( 1968 ) , 210 .
3) C. Caroli and K. Maki , Phys. Rev . 164 ( 1967 ), 591 .
4) H. Ebisawa and H. Takayama , Prog. Theor. Phys . 42 ( 1969 ), 1481 .
5) W. Weller , Phys. Stat. Solidi 35 ( 1969 ) , 573 .
6) H. Fukuyama , H. Ebisawa and T. Tsuzuki , Phys. Letters 33A ( 1970 ), 187 .
7) K. Maki , Physics 1 ( 1964 ), 21 .
8) L. P. Gor 'kov , Soviet Phys.JETP 10 ( 1960 ), 998 .
9) H. Takayama and H. Ebisawa , Prog. Theor. Phys . 44 ( 1970 ), 1450 .
10) R. Kubo , J. Phys. Soc. Japan 19 ( 1964 ), 2127 .
11) K. Maki , Prog. Theor. Phys . 39 ( 1968 ), 897 .
12) E. Abrahams and T. Tsuneto , Phys. Rev . 152 ( 1966 ), 416 .
13) K. Maki , Phys. Rev. Letters 23 ( 1969 ), 1223 .